2011 3rd Conference on Data Mining and Optimization (DMO) 28-29 June 2011, Selangor, Malaysia

978-1-61284-212-7/11/$26.002011 IEEE

Greedy Constructive Heuristic and Local Search Algorithm for Solving Nurse Rostering Problems

Rema A. Abobaker, Masri Ayob and Mohammed Hadwan Faculty of Information Science and Technology

Universiti Kebangsaan Malaysia 43600 Bangi, Selangor, Malaysia

Jawa2000d@yahoo.com, masri@ftsm.ukm.my and hadwan79@gmail.com

AbstractThe Nurse Rostering Problem (NRP) is one of the NP-hard problems that are difficult to solve for optimality. NRP deals with the distribution of working shifts to the staff nurses at healthcare organizations under given rules. Normally, NRP aims at generating a duty roster that satisfies all of the hard constraints (mandatory) and as many soft constraints (optional) as possible. This work introduces a greedy constructive heuristic algorithm, based on building two patterns of two-weeks duration that satisfies all of the hard constraints and several soft constraints. The first pattern is designed mainly to fulfill the night shift coverage whilst the second pattern is concerned with morning and evening shifts only. If the solution is not feasible (for example the coverage is not met), a repairing mechanism algorithm is applied until a feasible solution is reached. Simulated Annealing (SA) is then used to improve the constructed solution. A real dataset from Universiti Kebangsaan Malaysia Medical Center (UKMMC) is used in this work to test the proposed approach. Currently, the duty roster at UKMMC is still constructed manually. Promising results have been obtained and presented in this paper. Keywords: Nurse Rostering, Greedy Constructive Heuristic, Simulated Annealing and Scheduling.

I. INTRODUCTION

The Nurse Rostering Problem (NRP) is one the timetabling problems that are concerned with the distribution of the shift assignments among the available staff nurses to fairly fulfill the operational requirements for a particular period. Typically, the nurses are assigned to work one of the following shifts: (i) morning (M), (ii) evening (E), (iii) night (N) or (iv) day off (O).The distribution of these shifts mainly comply with the staff preferences, the policy laid down by the administration and the labor agreement clauses[1]. The complexities and the challenges of nurse rostering problems arise from the fact that a large variety of constraints exist, some of which contradict each other. The nurses duty roster is a timetable that consists of the shift assignments and rest days of staff nurses at healthcare organizations. Normally, hospitals are divided into several wards; each ward has its own staff nurses who are working at the same ward. The head nurse is the one who is responsible of producing and managing the duty rosters of each ward.

In the literature of timetabling, many approaches and methods have been used to solve NRP. Several overview articles have been published which summarized the progress in the area of nurse rostering, such as [2-4].For example [2], provides a thorough overview of proposed approaches and methods up to 2004. Based on the survey paper of [2], the proposed method is classified into, heuristics, meta-heuristics and artificial intelligence. Meta-heuristic approaches show a different rate of success in solving time tabling problems in general and NRP specifically[2]. Meta-heuristic approaches have been proposed to solve NRP through methodology such as variable neighborhood [5], tabu search [6], simulated annealing [7], genetic algorithm [8] and others [9, 10]. Heuristic approaches are usually considered as the fastest approximate methods that can produce solutions from scratch by adding opportunely defined solution components to an initially empty partial solution. This is accomplished until a solution is complete or other stopping criteria are fulfilled; this is known as a greedy approach[11].

In [12], a simulated annealing and a simple local search heuristic are merged to produce cyclical schedules for personnel in continuously operating organizations. A Simulated Annealing algorithm and Genetic algorithm are used to solve staff scheduling in[13]. A simulated annealing is also used in [7] to improve the initial solution that was constructed based on the shift patterns approach. In [7] they used a group of one-week feasible patterns to construct the nurse duty roster for a two-week duration, while in this work, we only used two patterns of two-week periods to fill up the time slots for each nurse.

The aim of this paper is to present our proposed constructive approach that uses a greedy shift pattern approach and simulated annealing to solve NRP at UKMMC.

II. UKMMC NURSE ROSTERING PROBLEM

In this paper we are solving the same problem in [7], as to solve the NRP at Universiti Kebangsaan Malaysia Medical Center(UKMMC). UKMMC is an educational hospital which belongs to the National University of Malaysia. The hospital has more than 1300 nurses attending about 900 beds. There are three shifts a day; two day-time shifts of 7 hours each (morning and evening) and a 10-hour long night

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shift. The nurses are also categorized as either senior or junior.

So far, the manual method scheduler is still used at UKMMC to generate fourteen-day duty rosters for their staff nurses. In fact, many hospitals still use a manual scheduler around the world which is a frustrating and time consuming task[2].

In [14], an exploration study is made of NRP at UKMMC and a mathematical model is presented, including hard and soft constraints and other related aspects to UKMMC NRP in their paper. For more details about UKMMC NRP refer to [7,14].

A. UKMMC Hard and Soft Constrains

Constraints in NRPs are usually classified into two groups: hard and soft constraints, which vary significantly with respect to official regulations and individual requirements, depending on individual organizations and countries [2]. Hard constraints have to be satisfied to get feasible solutions whilst soft constraints are desirable to be satisfied but not mandatorily, and thus can be violated [15]. The quality of the produced roster is dependent on the total penalty average; a lower total penalty indicates the better duty roster.

Hard constraints are: H1. Coverage of shifts must be at least for the demanded

number of nurses. H2. Nurses cannot work more than one shift a day. H3. A rest day must be included, with at least two days per

two-week period for every nurse. H4. At least one senior nurse is presented for every shift. H5. There must be no isolated working days. H6. The minimum working load is 10 days for each two-

week period and the maximum is 12 days. H7. The maximum span of consecutive working load is 4

days. H8. If the nurse has four consecutive night shifts this must

be followed by two rest days. The soft constraints are: S1. Attempt to give an equal number of working days and

rest days for every nurse. S2. Attempt to give each nurse at least one rest day in the

weekend during the two-week duty roster. S3. Attempt to give four consecutive morning shifts

followed by one day off. S4. Attempt to give four consecutive evening shifts followed

by one day off. S5. Attempt to give an evening shift after the rest days that

follow the night shift pattern. B. Coverage Demand Coverage demand is among the most important constraints that need to be satisfied. In the UKMMC datasets, the coverage demand is different from one shift to another and

from the weekdays to the weekends. Table 1 presents the coverage demand of the UKMMC datasets.

TABLE 1 THE MINIMUM COVERAGE DEMAND.

C. Objective function

Generally, the objective function is used to measure the quality of the produced duty roster. In this work, the objective function is a minimization-based method that attempts to reduce the total penalty occurring due to the violation of soft constraints as in [14]. Each soft constraint is weighted according to its importance. There is no standard weight for the soft constraints and as in [16], the weights of each soft constraint are set based on the discussion with the head nurses at UKMMC. Usually, the higher weight is used to indicate the higher value and importance of the specific soft constraints to be satisfied which are shown in table2.

TABLE2 THE WEIGHT OF EACH SOFT CONSTRAIN No. Soft constraints Weight of

violation S1 Give an equal number of working days and

rest days for every nurse. 100

S2 Give each nurse at least one rest day in the end of the week during the two-week.

100

S3 Give four consecutive morning shifts followed by one day off.

10

S4 Give four consecutive evening shifts followed by one day off.

10

S5 Give an evening shift after the rest day that follows night shift.

1

Table 3 shows an example of some shift sequences for the problem.

TABLE3 SAMPLE FOR SHIFT SEQUENCES Shift Sequence Penalty Comment NNNNOOE MMMMO EEMMO MMEEO NNNNOOM

0 0 10 10 1

Satisfy S5 Satisfy S4 M not allowed to follow E E not allowed to follow M M not allowed to follow days off that followed night shifts

Note: where M=morning shift, E=evening shift, N=night shift, O=day off. An example of calculating the total penalty for the soft constraint (S1) violations is presented in table 4 where two nurses have different working load.

TABLE 4. SAMPLE FOR CALCULATION OF S1 Nurse Total working

days Total No. of

days off Violation of S1

N1 11 3 100

Weekdays (Mon-Fri) Weekend (Sat-Sun) Dataset nurses Senior

nurses Morning Evening Night

Morning

Evening

Night

CICU 11 8 3 3 2 2 2 2 SGY5 18 11 4 4 3 4 4 2 MD1 19 12 4 4 3 4 4 2 N50 50 31 10 10 10 10 10 10

GCIU 73 38 16 16 15 15 15 14

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DN= number of minimum coverage demand for night shifts; CN= the maximum number for consecutive night shifts; Repeat Repeat Set the first Pattern; Rotate the first pattern CN positions Until DN is met Repeat Set the second pattern Rotate the second pattern one position Until (number of nurses is reached) To (number of nurses) Check Coverage: Repeat Repeat If (required coverage is not reached) (Swap rest day with working day) To (All days of the period) To ((requested coverage is reached) or (all nurses have been checked)) Repeat Repeat (Check hard constraints) (Take from extra shift to cover a missed shift) To (All days of the period) To (all nurses have been checked)

N2 10 4 0 Total penalty of S1 100

Note: If total working days

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Procedure SIMULATED_ANNEALING; Determine initial candidate solution s Set initial temperature T according to annealing schedule While termination condition not satisfied: Probabilistically choose a neighbor of s If satisfies probabilistic acceptance criterion (Depending on T): s := Update T according to annealing schedule

IV. SIMULATED ALGORITHM OPTIMISER

The Simulated Annealing (SA) algorithm is inspired by the process of solid material that is initially heated and then cooled in order to develop particular characteristics. In optimisation problems this principle is applied in order to avoid local optima in the search space. There must be an existing initial solution in order to apply SA to develop this initial solution. The SA procedure works by moving from the current solution candidate to another neighbor candidate based on some criterion that accepts this move if it develops the current objective value or leaves it without any change based on predefined moves. At this stage, we are trying to reach the global optimal solution which is the best duty roster found so far in this work. In SA, a solution in the neighborhood is chosen randomly, and then evaluated. Three neighborhood moves are used in this work such as in [17-20]. These moves are: 1. Swap single shift between two days of a single nurse

e.g. a day off and a day worked were swapped, thus not affecting the cover constraints but potentially affecting the other constraints and objectives.

2. Swapping of single shift between two nurses on the same day e.g. evening shift is swapped by morning shift.

3. Change of single shift to another single shift type of a single nurse, for example a working day is changed to a day off. Refer to figure 4 for the simulated annealing procedure.

FIGURE 4. THE SIMULATED ANNEALING PROCEDURE The procedure of simulated annealing is the standard simulated annealing as in[21].

V. EXPERIMENTAL RESULT

For the parameter settings that are used in this work [22], temperature t=2000, cooling rate =o.98, 0.001with probability function =/, stopping criteria are to meet t

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The authors wish to thank everyone who has helped in this research.

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[15] E. Burke, H. Rudov, and E. zcan, "Memes, Self-generation and Nurse Rostering," in Practice and Theory of Automated Timetabling VI. vol. 3867: Springer Berlin / Heidelberg, 2007, pp. 85-104.

[16] E. K. Burke, T. Curtois, G. Post, R. Qu, and B. Veltman, "A Hybrid Heuristic Ordering and Variable Neighbourhood Search for the Nurse Rostering Problem," European Journal of Operational Research, vol. 188, pp. 330-341, 2008.

[17] A. Jaszkiewicz, " A metaheuristic approach to multiple objective nurse scheduling," Foundations of Computing and Decision Sciences, vol. 22, pp. 169-183, 1997.

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[22] S. Kundu, M. Mahato, B. Mahanty, and S. Acharyya, "Comparative Performance of Simulated Annealing and Genetic Algorithm in Solving Nurse Scheduling Problem," in Proceedings of the International MultiConference of Engineers and Computer Scientists 2008, Hong Kong, 2008.

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